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Finding derivative question

  1. Oct 4, 2006 #1
    Finding derivative question plz....

    How should I find the derivative of y = x^(e^x)?

    I tried using the chain rule along with the power rule, coming out to:
    (e^x) (e^x) (X^(e^x - 1))

    If I had took the natural log of both sides and then used implicit differentiation, I would have gotten as a derivative:
    (x^(e^x)) (e^x) (1/x + ln x)
    which is the correct answer according to my TI89.

    Why wouldn't the first method work? Or was there any flaw?

    By the way I just started Calculus as a high schooler.
  2. jcsd
  3. Oct 4, 2006 #2
    The power rule only works for functions of the form [tex] f(x) = x^{n} [/tex]. So you can't use it for functions in the form of [tex] f(x) = x^{g(x)} [/tex] You could use implicit differentiation. Or you could do the following:

    [tex] x^{e^{x}} = (e^{\ln x})^{e^{x} [/tex].

    [tex] \frac{d}{dx} ( x^{e^{x}})= \frac{d}{dx}(e^{\ln x}^{e^{x}}) = e^{\ln xe^{x}}\frac{d}{dx}(e^{x}\ln x) [/tex]

    [tex] \frac{dy}{dx} = e^{\ln x e^{x}}(\frac{e^{x}}{x}+e^{x}\ln x) [/tex]

    [tex] \frac{dy}{dx} = x^{e^{x}}e^{x}(\frac{1}{x}+ \ln x) [/tex]

    Note: [tex] e^{\ln x e^{x}} = (e^{\ln x})^{e^{x}} = x^{e^{x}} [/tex]
    Last edited: Oct 4, 2006
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